Let's solve the equation step-by-step:
1. Rewrite the equation for clarity:
[tex]\[
2x^2 - 1 = -x
\][/tex]
2. Move all terms to one side of the equation to set it to zero:
[tex]\[
2x^2 + x - 1 = 0
\][/tex]
3. Factor the quadratic equation:
To factor [tex]\(2x^2 + x - 1\)[/tex], we look for two numbers that multiply to [tex]\(2 \cdot (-1) = -2\)[/tex] and add to [tex]\(1\)[/tex]. These numbers are [tex]\(2\)[/tex] and [tex]\(-1\)[/tex].
Rewriting the middle term using these numbers:
[tex]\[
2x^2 + 2x - x - 1 = 0
\][/tex]
Group and factor by grouping:
[tex]\[
(2x^2 + 2x) - (x + 1) = 0
\][/tex]
[tex]\[
2x(x + 1) - 1(x + 1) = 0
\][/tex]
4. Factor out the common binomial factor:
[tex]\[
(2x - 1)(x + 1) = 0
\][/tex]
5. Set each factor to zero and solve for [tex]\(x\)[/tex]:
[tex]\[
2x - 1 = 0 \quad \text{or} \quad x + 1 = 0
\][/tex]
6. Solve each equation:
[tex]\[
2x - 1 = 0 \implies 2x = 1 \implies x = \frac{1}{2}
\][/tex]
[tex]\[
x + 1 = 0 \implies x = -1
\][/tex]
The solutions to the equation are:
[tex]\[
x = -1 \quad \text{and} \quad x = \frac{1}{2}
\][/tex]
Therefore, the solutions are [tex]\(x = -1, \frac{1}{2}\)[/tex].
